STAY IN PHASE

by

John Ehlers

Source: MESA Software Technical Papers

 

INTRODUCTION

A Cycle is the one market characteristic that can be scientifically measured. Although they can be measured they are maddening because they are ephemeral - they come and go. Our recent research shows there is a fundamental cycle parameter that leads us to the correct trading strategy for any current market mode. We start by defining two possible market modes. These modes are the Trend Mode and the Cycle Mode. In the Trend Mode the correct strategy is to buy (or sell, for down trends) and hold. In the Cycle Mode the correct strategy is to buy and sell on the cyclic valleys and peaks.

The parameter we use is the phase of the cycle. The measured phase tells us with great sensitivity when we are in the Trend Mode, enabling the capture of a large fraction of the trend movement. This capture range is typically far larger than can be obtained with crossing moving average or other typical trend identification techniques. In the Cycle Mode the measured phase pinpoints the cyclic turns IN ADVANCE, with the further advantage that the false whipsaw signals of typical "oscillator" type signals are avoided.

 

THE NATURE OF PHASE

To use phase, we must first understand what it is. It, quite simply, is a description of where we are in the cycle. Are we at the beginning, middle, or end of the cycle? Phase is a quantitative description of that location. Each cycle passes through 360 degrees to complete the cycle. One basic definition of a cycle is that it consists of an action having a uniform rate-change of phase. For example, a 10 day cycle passes through 360 degrees every 10 days. For a perfect cycle it must change phase at the rate of 36 degrees per day each day throughout the cycle.

How does this help us see a Trend Mode? Easy. By reverse logic. In a Trend Mode there is no cycle, or at least a very weak one. Therefore there is no rate change of phase. So, if we compare the rate change of measured phase to the theoretical rate change of phase of the weak dominant cycle present in the Trend Mode, we get a correlation failure. This failure to correlate the two cases of the rate change of phase enables us to define the presence of a trend. Knowing we have a trend, it is easy to set our strategy to a simple buy-and-hold until the trend disappears.

 

Figure 1. Phase describes the location within a cycle

 

Figure 2. A sinewave in the time domain can be generated by placing a pen on the arrowhead and drawing the paper along at a uniform rate, just like a seismograph.

 

Figure 3. Phase varies uniformly throughout the cycle, and is drawn as reset to show the beginning of a new cycle

One easy way to picture a cycle is as an indicator arrow bolted to a rotating shaft as shown in the phasor diagram of Figure 1. Each time the arrowhead sweeps through one complete rotation a cycle is completed. The phase increases uniformly throughout the cycle as shown in Figure 2. The phase continues on for the next cycle, but is usually drawn as being reset to zero to start the next cycle. If we additionally place a pen on the arrowhead and draw a sheet of paper below the arrowhead at a uniform rate, like they do for seismographs, the pen draws a theoretical sinewave. The relationship between the phasor diagram and the theoretical sinewave is shown in Figure 3. The sinewave is the typical cycle waveform we recognize in the time domain on our charts. The phase angle of the arrow uniquely describes where we are in the time domain waveform.

The position of the tip of the arrow in Figure 1 can be described in terms of the length of the arrow, L, and the phase angle, q. If we let the arrow be the hypotenuse of a right triangle we can convert the description of the arrow from length and angle to two orthogonal components - the other two legs of the right triangle. The vertical component is L*Sin(q) and the horizontal component is L*Cos(q). The ratio of these two components is the tangent of the phase angle. So, if we know the two components, all we have to do to find the phase angle is to take the arctangent of their ratio. This is something that may be tough for you, but it's a piece of cake for your computer.

We measure the phase of the dominant cycle by establishing the average lengths of the two orthogonal components. This is done by correlating the data over one fully cycle period against the sine and cosine functions. Once the two orthogonal components are measured, the phase angle is established by taking the tangent of their ratio. A simple test is to assume the price function is a perfect sinewave, or Sin(q). The vertical component would be Sin²(q) = .5*(1-Cos(2q)) taken over the full cycle. The Cos(2q) term averages to zero, with the result that the correlation has an amplitude of Pi. The horizontal component is Sin(q)*Cos(q) = .5*Sin(2q). This term averages to zero over the full cycle, with the result that there is no horizontal component. The ratio of the two components goes to infinity because we are dividing by zero, and the arctangent is therefore 90 degrees. This means the arrow is pointing straight up, right at the peak of the sinewave.

One additional step in our calculations is required to clear the ambiguity of the tangent function. In the first quadrant both the sine and cosine have positive polarity. In the second quadrant the sine is positive and the cosine is negative. In the third quadrant both are negative. Finally, in the fourth quadrant the sine is negative and the cosine is positive. The phase angle is obtained regardless of the amplitude of the cycle. Given that we know the dominant cycle, the BASIC program in the sidebar shows how we can compute the phase angle.

 

PUTTING THE PHASE TO WORK

Figure 4a. Plotting the sine of the measured phase angle

Figure 4b. Sinewave Indicator is created by advancing the measured phase by 45 degrees.

We can make an outstanding cyclic indicator simply by plotting the Sine of the measured phase angle. When we are in a Cycle Mode this indicator looks very much like a sinewave. When we are in a Trend Mode the Sine of the measured phase angle tends to wander around slowly because there is only an incidental rate change of phase. A clear, unequivocal indicator can be generated by plotting the Sine of the measured phase angle advanced by 45 degrees. This case is depicted for the phasor diagram and the time domain in Figure 4b. The two lines cross SHORTLY BEFORE the peaks and valleys of the cyclic turning points, enabling you to make your trading decision in time to profit from the entire amplitude swing of the cycle. A significant additional advantage is that the two indicator lines don't cross except at cyclic turning points, avoiding the false whipsaw signals of most "oscillators" when the market is in a Trend Mode. The two lines don't cross because the phase rate of change is nearly zero in a trend mode. Since the phase is not changing, the two lines separated by 45 degrees in phase never get the opportunity to cross.

If the rate of change of the measured phase does not correlate with the theoretical phase rate-change of the dominant cycle, then a Trend must be in force. A workable definition is a Trend exists when the measured phase rate of change is less than 67% of the theoretical phase rate of the dominant cycle. This is a very sensitive detector for the Trend Mode, enabling you to capture high percentages of the Trend movement.

 

REAL WORLD EXAMPLES

Theory is nice, and is always how I initially attack each trading problem. It is possible for theory to be a laboratory curiosity with no practical application. Happily, this is not the case for the use of phase in trading.

 

Figure 5. Cycle and Phase Response of September 1996 Deutschemark

Figure 5 is a MESA96 display for the September 96 Deutschemark contract. The price bars are displayed in the top segment with two adaptive moving average overlays. The second segment is the Sinewave Indicator, plots of the Sine of the measured phase angle and the phase angle advanced by 45 degrees. The measured phase is displayed below the Sinewave Indicator, and the measured dominant cycle and spectrum are displayed in the bottom segment.

From the way the phase varied uniformly in March, it is clear that DM was in the Cycle Mode during that month. As a result, the Sinewave Indicator looks like a sinewave and gives two buy and one sell signal where the Sinewave Indicator lines cross.

The phase stopped changing at a uniform rate in April and May because two cycles identified by the spectrum display were present simultaneously. Since the phase hardly changed from day to day, DM went into a Trend Mode during these two months. The trend mode is identified, in TradeStation lingo, by "PaintBars", where the price bars are violet. The correct trading strategy during April and May was to hold a short position (a move worth over $2500 per contract) because the faster adaptive moving average was below the slower one. Another way to identify the downtrend is that the phase remained near zero degrees (or near 360 degrees) during these two months. Notice that the Sinewave Indicator does not give false whipsaw signals during April and May. Whipsaws in the Trend Mode are common for "oscillator" indicators such as the Stochastic and RSI.

The phase resumed its uniform rate of change during June and into July because the dominant cycle settled down to a relatively stationary value. As a result, the Cycle Mode appeared, the Sinewave Indicator again looks like a sinewave, and four excellent sell signals and three excellent buy signals resulted.

 

Figure 6. Cycle and Phase Response of September 1996 Treasury Bonds

Figure 6, showing the cycle and phase response of the September contract of US Treasury Bonds, is another example of how phase can be used to sharpen your trading. In February and March, Bonds were in a decline. The phase hovered near zero degrees, clearly identifying the downtrend. The correct trading strategy during this period was to hold a short position until the first cyclic buy signal given by the Sinewave Indicator early in April. From that first cyclic buy signal there were three more cyclic buy signals and three cyclic sell signals in the ensuing three months. Bonds didn't stay exclusively in the Cycle Mode during all that time because the cycle length tended to wander around. However, the cycle wandering only introduced distortions in the shape of the Sinewave Indicator. The crossover signals it produced unequivocal and would have produced substantial profits in every case.

 

CONCLUSION

Phase is an exciting new parameter to help technicians analyze the market. It can help in several regards. First, it enables you to establish your trading strategy to fit the Trend Mode in the Cycle Mode. If you are not comfortable trading the Cycle Mode, you can always stand aside for a while until a new Trend Mode is established. If you want to trade the Cycle Mode, the Sinewave Indicator, created by plotting the Sine of the phase angle and the Sine of the phase angle advanced by 45 degrees, gives clear buy and sell signals in advance of each cyclic turn. Getting the signal in advance enables you to make your entry and exit right at the cyclic turning point without giving up a piece of the market movement. The measured phase, Trend Mode identification, and Sinewave Indicator are all part of the MESA96 trading software program.

 

 

 

SIDEBAR

BASIC Code for Phase Calculation

 

This BASIC code finds the real part of the phasor (the horizontal component) and the imaginary part of the phasor (the vertical component) by summing the product of the price and the two sinusoids over one full cycle of the dominant cycle. The arctangent function locates the phase to be in the first or fourth quadrant. The quadrant ambiguity is removed by adding p to the phase angle is the real part is negative. A value of p/2 is arbitrarily added to the computed phase so the start of the cycle is referenced to a Sine wave. The computed phase angle is then tested to fall within the range from 0 to 2p. The phase is then converted to degrees from radian measure.

An interesting observation is that if the price is a linear slope, summing the product of the price and a sine over a cycle is the discrete equivalent of the integral ò x Sin(x) dx. Correspondingly, the real part is the equivalent of the integral ò x Cos(x) dx. Working through these theoretical examples, we find that the phase is 180 degrees for a trending upslope and is zero degrees for a trending downslope. Thus, phase can be a sensitive way to detect a Trend.